一种用于空间分数微分方程的有限差分方法（变系数）
A finite difference method for space fractional differential equations with variable diffusivity coefficient
AI总结：
本文提出并分析了一种用于求解变系数分数微分方程的有限差分方法，结合了向前和向后欧拉方法近似双侧分数导数，证明了方案的解存在性和唯一性，并展示了截断误差为O(h)的精度。
摘要：
异常扩散现象无法用二阶扩散方程准确建模，但更适用于分数扩散模型。分数扩散算子的非局部性质使这些模型的数学分析和数值方案的建立变得更加困难。本文提出并分析了用于求解一维空间中变系数分数微分方程的有限差分方法，该方法结合了一阶向前和向后欧拉方法近似左侧分数导数，当右侧分数导数用两次一阶向后欧拉方法应用时进行近似。所提出的差分方案在无分数导数时退化为标准二阶中心差分方案。证明了所提方案的解的存在性和唯一性，并展示了截断误差为O(h)的精度，其中h表示最大空间步长。数值测试展示了该方案的全局O(h)精度，除了非光滑情况外，非光滑情况如预期般导致收敛率下降。
英文摘要：
Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving {\em variable-coefficient} fractional differential equations, with two-sided fractional derivatives, in one-dimensional space. The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided fractional derivative when the right-sided fractional derivative is approximated by two consecutive applications of the first-order backward Euler method. Our finite difference scheme reduces to the standard second-order central difference scheme in the absence of fractional derivatives. The existence and uniqueness of the solution for the proposed scheme are proved, and truncation errors of order $h$ are demonstrated, where $h$ denotes the maximum space step size. The numerical tests illustrate the global $O(h)$ accuracy of our scheme, except for nonsmooth cases which, as expected, have deteriorated convergence rates.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
